$2$ Time Delay Margin

Consider the controller $C(s)=\frac{4}{s}$ that has been designed for the nominal plant $P^o(s)=2$ in a standard feedback system with negative feedback.

$($a$)$ Find the closed$-$loop transfer function from $r$ to $y.$ Is the closed$-$loop system stable?

$($b$)$ Find the steady$-$state gain from $ry.$

$($c$)$ What is the time$-$constant of the closed$-$loop system?

$($d$)$ What is the phase margin of the system?

$($e$)$ How much time delay can the system tolerate before becoming unstable? $($This is called the "time$-$delay margin"$).$ Denote it by $T_d.$

$($f$)$ Verify your answers with Simulink. Plot the unit step response of the closed$-$loop system with the following time delays: $0,\frac{1}{10}{T}_d,\frac{3}{10}{T}_d,\frac{6}{10}{T}_d,\frac{9}{10}{T}_d,\frac{9\mathrm{.}9}{10}{T}_d.$ You should notice oscillations of increasing magnitudes.